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An independent set in a graph G is a set of pairwise nonadjacent vertices, and the maximum size, alpha(G), of an independent set in G is called the independence number. Given a graph G and weight matrix A of G with entries from some field F, the maximum dimension of an A-isotropic subspace, known as the Witt index of A, is an upper bound on alpha(G). Since any weight matrix can be used, it is natural to seek the minimum upper bound on the independence number of G that can be achieved by a weight matrix. This minimum, iota_F^*(G), is called the minimum Witt index of G over F, and the resulting bound, alpha(G)<= iota_F^*(G), is called the isotropic bound. When F is finite, the possible values of iota_F^*(G) are determined and the graphs that attain the isotropic bound are characterized. The characterization is given in terms of graph classes CC(n,t,c) and CK(n,t,k) constructed from certain spanning subgraphs called C(n,t,c)-graphs and K(n,t,k)-graphs. Here t is the term rank of the adjacency matrix of G. When F=R, the isotropic bound is known as the Cvetkovi\'c bound. It is shown that it is sufficient to consider a finite number of weight matrices A when determining iota_R^*(G) and that, in many cases, two weight values suffice. For example, if the vertex set of G can be covered by alpha(G) cliques, then G attains the Cvetkovi\'c bound with a weight matrix with two weight values. Inequalities on alpha and iota_F^* resulting from graph operations such as sums, products, vertex deletion, and vertex identification are examined and, in some cases, conditions that imply equality are proved. The equalities imply that the problem of determining whether or not alpha(G)=iota_F^*(G) can be reduced to that of determining iota_F^*(H) for certain crucial graphs H found from G. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2007-09-04 15:38:47.57

Multiple-Input Multiple-Output (MIMO) communications systems using multiantenna arrays simultaneously during transmission and reception have generated significant interest in recent years. Theoretical work in the mid 1990?s showed the potential for significant capacity increases in wireless channels via spatial multiplexing with sparse antenna arrays and rich scattering environments. However, in reality the capacity is significantly reduced when the antennas are placed close together, or the scattering environment is sparse, causing the signals received by different antennas to become correlated, corresponding to a reduction of the effective number of sub-channels between transmit and receive antennas. By introducing the previously ignored spatial aspects, namely the antenna array geometry and the scattering environment, into a novel channel model new bounds and fundamental limitations to MIMO capacity are derived for spatially constrained, or spatially selective, channels. A theoretically derived capacity saturation point is shown to exist for spatially selective MIMO channels, at which there is no capacity growth with increasing numbers of antennas. Furthermore, it is shown that this saturation point is dependent on the shape, size and orientation of the spatial volumes containing the antenna arrays along with the properties of the scattering environment. This result leads to the definition of an intrinsic capacity between separate spatial volumes in a continuous scattering environment, which is an upper limit to communication between the volumes that can not be increased with increasing numbers of antennas within. It is shown that there exists a fundamental limit to the information theoretic capacity between two continuous volumes in space, where using antenna arrays is simply one choice of implementation of a more general spatial signal processing underlying all wireless communication systems.

In this thesis, based on the Onsager excluded volume interaction model, two nematic ordering problems of wormlike (semiflexible) polymer are studied: one is to investigate the isotropic-nematic interface of polymers for three typical cases&mdash;the flexible one, the rigid-rod one and the intermediate one; the other is to investigate a very long polymer confined between two infinite flat hard walls. <br /><br /> Many previous studies of the isotropic-nematic phase coexistence are mainly focused on either rigid rod-like polymers with small flexibility, or flexible polymers with large flexibility. The phase coexistence of polymers with intermediate flexibility is desired to be investigated. For these three typical cases (flexible, rigid-rod and intermediate), the profiles for density, order parameter and tension contribution were shown for different tilt angles. The interface tension was studied. The simulation results are consistent with those reported by other people. <br /><br /> We investigated the confinement of a long polymer between two flat hard walls, which are separated by a distance comparable to the effective Kuhn length of polymer chain by the wormlike chain model with or without the Onsager excluded volume interaction. Without the interaction, the results are compared with those of the Gaussian chain model. Including the interaction, the phase diagram is analyzed.

A semiflexible polymer confined to a spherical surface is used as a basic model for understanding DNA conformation in restricted space. By means of Monte Carlo simulation for a bead-rod chain generated on a spherical surface, we find an ordered helicoidal phase at sufficiently high surface density and determine the critical density of the isotropic-helicoidal phase transition for various persistence lengths. We verify that the excluded volume effect is the key factor to cause the helicoidal state. In addition to Monte Carlo simulations, we utilize the model of wormlike chain with Onsager's excluded volume interaction and examine the Landau expansion of the free energy involving both the orientational and spatial order parameters. We also analytically figure out the critical density and transition gap for various ratios of persistence lengths of the polymer chain and the radius of spherical surface. The results from both simulation and analysis are consistent with each other.

In this thesis, based on the Onsager excluded volume interaction model, two nematic ordering problems of wormlike (semiflexible) polymer are studied: one is to investigate the isotropic-nematic interface of polymers for three typical cases&mdash;the flexible one, the rigid-rod one and the intermediate one; the other is to investigate a very long polymer confined between two infinite flat hard walls. <br /><br /> Many previous studies of the isotropic-nematic phase coexistence are mainly focused on either rigid rod-like polymers with small flexibility, or flexible polymers with large flexibility. The phase coexistence of polymers with intermediate flexibility is desired to be investigated. For these three typical cases (flexible, rigid-rod and intermediate), the profiles for density, order parameter and tension contribution were shown for different tilt angles. The interface tension was studied. The simulation results are consistent with those reported by other people. <br /><br /> We investigated the confinement of a long polymer between two flat hard walls, which are separated by a distance comparable to the effective Kuhn length of polymer chain by the wormlike chain model with or without the Onsager excluded volume interaction. Without the interaction, the results are compared with those of the Gaussian chain model. Including the interaction, the phase diagram is analyzed.

A semiflexible polymer confined to a spherical surface is used as a basic model for understanding DNA conformation in restricted space. By means of Monte Carlo simulation for a bead-rod chain generated on a spherical surface, we find an ordered helicoidal phase at sufficiently high surface density and determine the critical density of the isotropic-helicoidal phase transition for various persistence lengths. We verify that the excluded volume effect is the key factor to cause the helicoidal state. In addition to Monte Carlo simulations, we utilize the model of wormlike chain with Onsager's excluded volume interaction and examine the Landau expansion of the free energy involving both the orientational and spatial order parameters. We also analytically figure out the critical density and transition gap for various ratios of persistence lengths of the polymer chain and the radius of spherical surface. The results from both simulation and analysis are consistent with each other.

The adhesive contact problems of a conical frustum punch indenting a transversely isotropic elastic half space and an orthotropic elastic half space are analytically studied in this thesis work. To solve the problem involving a transversely isotropic half space, the harmonic potential function method and the Hankel transform are employed, which lead to a general closed-form solution for the adhesive contact problem. For the case with an orthotropic half space, the problem of a point load applied on the half space is first solved by using the double Fourier transform method. The solution for the adhesive contact problem is then obtained through integrating the former solutions over the punch surface.

This study is to investigate that the polishing behavior will be independent of or dependent on the direction of particle motion by the anisotropic polishing phenomenon of hydrodynamic polishing process under the semi-contact lubricating condition. There are two types of experiments to be examined to get to the objectives. First, taking polishing on the work surface which possesses the isotropic surface roughness, to discuss the variation of the smoothing efficiency of the surface irregularities in the five different directions on the work surface. Second, taking three kinds of polishing, ¡§longitudinal, transverse, and oblique roughness polishing¡¨, on the work surface which possesses the anisotropic surface roughness. Then to discuss the variation of the smoothing efficiency of the surface irregularities on the work surface. Both the results of two types experiments should be take to distinguish the difference between one smoothing efficiency and the others from using the hypothesis testing. All hypothesis tests about the experiment results of the work piece which possesses the isotropic surface roughness are accepting . But, most hypothesis tests about the experiment results of the work piece which possesses the anisotropic surface roughness are rejecting . The theory analysis about the smoothing efficiency is discussing. The discussion about the smoothing efficiency can explain the phenomenon due to taking polishing on the work surface which possesses the anisotropic surface roughness. The reason why the phenomenon happened is possible the effects of different lubrication condition. Last, from the lubrication theory, the effects of different lubrication condition due to different surface texture can be employed to verify the explanation about the phenomenon is suitable. The conclusion from the experiment results and the theory analysis is: the polishing behavior is possible independent of the direction of particle motion by the anisotropic polishing phenomenon of hydro- dynamic polishing process under the semi-contact lubricating condition.